End-periodic homeomorphisms and volumes of mapping tori

TL;DR

This paper establishes an upper bound on the minimal hyperbolic volume of certain 3-manifolds derived from end-periodic surface homeomorphisms, linking it to the translation length on the pants graph, and demonstrates the bound's sharpness through examples.

Contribution

It introduces a new volume bound for mapping tori of end-periodic homeomorphisms, extending Brock and Agol's finite-type results to infinite-type surfaces.

Findings

Derived an upper bound on hyperbolic volume based on translation length.

Constructed examples showing the bound is asymptotically sharp.

Connected the geometry of 3-manifolds with combinatorial properties of surface homeomorphisms.

Abstract

Given an irreducible, end-periodic homeomorphism f of a surface S with finitely many ends, all accumulated by genus, the mapping torus is the interior of a compact, irreducible, atoroidal 3-manifold with incompressible boundary. Our main result is an upper bound on the infimal hyperbolic volume of the compactified mapping torus in terms of the translation length of f on the pants graph of S. This builds on work of Brock and Agol in the finite-type setting. We also construct a broad class of examples of irreducible, end-periodic homeomorphisms and use them to show that our bound is asymptotically sharp.